We have checked the identity from Theorem REF  with a computer for all (not just coprime to \(n, n+1\) ) odd \(5<\ell <30\)  and \(1<n<10\)  and expect it to be true in general.


By the rational shuffle conjecture, resulting from e.g. [1]}, we know the bigraded scalar products \(\langle P_{m/n}\cdot 1, h_\lambda \rangle \)  for the \(m/n\) -case as well. It would be interesting to understand a (bi)graded version of Theorem REF  as well.
